Applications of definite integral

9000100004

Level: 
B
The function \(f(x)= x^{2} + 2\) is graphed in the picture. Consider the region bounded by the graph of the function, both axes and the line \(x = -1\). Determine the solid of revolution obtained by revolving this region about \(x\)-axis.
A general solid which is neither cone nor cylinder.
Cone with base of radius \(1\).
Cylinder with base of radius \(2\).
Cone with base of radius \(2\).

9000100005

Level: 
B
The function \(f(x) = 1\) is graphed in the picture. Determine the solid of revolution with volume given by the following formula. \[ \pi \int _{-1}^{1}f^{2}(x)\, \mathrm{d}x \]
Cylinder of base radius \(1\) and height \(2\).
Cone of base radius \(1\) and height \(2\).
Cone of base radius \(2\) and height \(1\).
Cylinder of base radius \(2\) and height \(1\).

9000100006

Level: 
B
The function \(f(x)= \sqrt{x}\) is graphed in the picture. Consider the region bounded by the graph of \(f\) on \([ 1;\, 4] \), lines \(x = 1\), \(x = 4\) and the \(x\)-axis. Identify the formula for volume of the solid of revolution obtained by revolving this region about the \(x\)-axis.
\(V =\pi \int _{ 1}^{4}x\, \mathrm{d}x\)
\(V =\int _{ 1}^{4}x\, \mathrm{d}x\)
\(V =\pi \int _{ 1}^{4}\sqrt{x}\, \mathrm{d}x\)
\(V =\int _{ 1}^{4}\sqrt{x}\, \mathrm{d}x\)

9000100007

Level: 
B
The function \(f(x)= \sqrt{x}\) is graphed in the picture. Consider the region bounded by the graph of \(f\) on \([ 1;\, 4] \), lines \(x = 1\), \(x = 4\) and the \(x\)-axis. Find the volume of the solid of revolution obtained by revolving this region about the \(x\)-axis.
\(\frac{15} {2} \pi \)
\(\frac{17} {2} \pi \)
\(\frac{17} {2} \pi ^{2}\)
\(\frac{15} {2} \pi ^{2}\)

9000100008

Level: 
B
Part of the graph of the function \(f(x) = \frac{1} {x}\) is shown in the picture. Complete the following sentence: „Formula \[ V =\pi \int _{ 1}^{2}x^{-2}\, \mathrm{d}x \] determines the volume of the solid of revolution obtained by revolving region bounded by ...”
\(x\)-axis, graph of \(f\) on \([ 1;\, 2] \) and lines \(x = 1\), \(x = 2\) about \(x\)-axis.
\(y\)-axis, graph of \(f\) on \([ 1;\, 2] \) and lines \(y = 1\), \(y = \frac{1} {2}\) about \(x\)-axis.
\(x\)-axis, graph of \(f^{2}\) on \([ 1;\, 2] \) and lines \(x = 1\), \(x = 2\) about \(x\)-axis.
\(y\)-axis, graph of \(f^{2}\) on \([ 1;\, 2] \) and lines \(y = 1\), \(y = \frac{1} {2}\) about \(x\)-axis.

9000100009

Level: 
B
Part of the graph of the function \(f(x) = \frac{1} {x}\) is shown in the picture. Consider the region bounded by \(x\)-axis, graph of \(f\) and lines \(x = 1\) and \(x = 4\). Find the volume of the solid of revolution obtained by revolving this region about \(x\)-axis.
\(\frac{3} {4}\pi \)
\(\frac{5} {4}\pi \)
\(\frac{5} {3}\pi \)
\(\frac{4} {3}\pi \)